On topological properties of families of finite sets

We present results about the Cantor–Bendixson index of some subspaces of a uniform family F of finite subsets of natural numbers with respect to the lexicographic order topology. As a corollary of our results we get that for any ω-uniform family F the restriction F↾M is homeomorphic to F iff M contains intervals of arbitrary length of consecutive integers. We show the connection of these results with a topological partition problem of uniform families.